l> Spectroscopic Notation A inside wall on Quantum Numbers and Spectroscopic Notation

### Contents

Quantum NumbersQuantum Numbers for AtomsSpectroscopic NotationTerms, Configurations, and also LevelsSelection RulesMultiplets M Degeneracy and the Zeeman impact Hyperfine StructureReferencesRegular patterns in the spectra of hydrogen (e.g., the Balmer series)and alkali steels lead Bohr topropose a design for the atom consist of of a positively fee nucleusorbited by electrons with discrete, quantized values of energy. When theBohr model is strictly applicable just to the most basic ions, those havinga single electron, the Bohr model listed enormous insights into thestructure the atoms, and also lead to the advance of quantum mechanics.Spectroscopists of the so late 19th and early 20thcentury created a device ofspectroscopic notation to describe the observed heat spectra. Quantum number were created to to administer an quantitative description ofobserved (and unobserved) transitions. This together carry out a short-hand summary of thestate that the electrons inan atom or ion (I will use the terms interchangeably). The notationis confusing, is case-specific, and sometimes ambiguous.

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## Quantum Numbers

4 quantum numbers suffice to describe any type of electron in an atom. This are: n
, the primary quantum number. This is additionally known as the radial quantum number, and defines the street of the electron native the cell core in the Bohr model. n additionally describes the azimuthal angular momentum. n takes on integral worths 1, 2, 3, ... . l, the azimuthal quantum number. In Sommerfeld"s generalization that the Bohr model, the one orbits that the electron are changed by elliptical orbits, and l explains the shape of the orbit. l bring away on the integral values 0, 1, 2, ... , n-2, n-1. If n=1, l=0. l is sometimes dubbed the reduced azimuthal quantum number, since the Sommerfeld formulation offered a quantum number k, which equals l+1. k=0 corresponds to no angular momentum, or a radial orbit i m sorry takes the electron v the nucleus. This is unphysical, and also is forbidden.m, the magnetic quantum number. m takes on the integral values -l , -(l-1), ..., -1, 0, 1, ..., (l-1), l. In Sommerfeld"s formulation, m explained the orientation that the ellipse. This is recognized as the magnetic quantum number due to the fact that its results are typically observed just under the influence of strong magnetic areas (which set a desired spatial orientation).s, the rotate quantum number. This defines the spin of the electron, and is either +1/2 or -1/2.The Pauli exemption principle needs that no two electronshave the same set of the 4 quantum number n, l,m,s, thereforethere are 2n2 feasible states for an electron with major quantum number n.The n=1 levels can contain just 2 electrons. This level is calledthe 1s orbit or the K shell (shells with n=1,2,3,4,5,6,7are referred to as the K, L, M, N, O, P, Q shells, respectively).An orbit, or shell, comprise themaximum number 2n2 electrons creates a closed shell.For example, once n=1, l=m=0 and also m=+/- 1/2.The feasible combinations that quantum numbers space nlms = 1, 0, 0,+1/2 and also 1, 0, 0, -1/2.When n=3, the 18 possible states are: 3, 0, 0, +1/2 and also 3, 0, 0, -1/2 3, 1, -1, +1/2 and also 3, 1, -1, -1/2 3, 1, 0, +1/2 and 3, 1, 0, -1/2 3, 1, 1, +1/2 and 3, 1, 1, -1/2 3, 2, -2, +1/2 and 3, 2, -2, -1/2 3, 2, -1, +1/2 and also 3, 2, -1, -1/2 3, 2, 0, +1/2 and 3, 2, 0, -1/2 3, 2, 1, +1/2 and 3, 2, 1, -1/2 3, 2, 2, +1/2 and also 3, 2, 2, -1/2Consequently, the statistical weightgn = 2n2 for the levelcharacterized by major quantum number n.In a hydrogenic atom or ion (H, the II, Li III, ... ), the power levels are fullydescribed through the rule quantum number n (aside from the finestructure terms, which modify the level by regards to order α,where the good structure continuous α is 1/137,and relativistic corrections needed for heavy ions). In a hydrogenic atom or ion, the frequency the a change between upperstate n=U and lower state n=L is ν = R Z2(1/(U2) - 1/(L2)) whereR is the Rydberg constant, 2π2με4/h2,where μis the lessened mass Mme/(M+me), M is the massive of thenucleus,me is the fixed of the electron,ε is the unit of electric charge,and h is Planck"s constant. R=13.598 eV. In the Bohratom, all electrons through commonn room degenerate, i.e., they every lie in ~ the same energy.

## Quantum Numbers for Atoms

As through electrons, 4 quantum numbers suffice to define the digital stateof an atom or ion.L is the complete orbital angular momentum.For hydrogenic ions and alkalis,with a solitary electron in the outer shell, L=l.L correspondsto the hatchet of the ion (S terms have actually L=0, ns terms haveL=1, etc.). In the situation of much more than one electron in the external shell,the value of L takes on all feasible values ofΣli(see Table 1, which is Table 5 from Herzberg). Table 1ElectronLTermConfigurations0Ssp1Ppp0 1 2S ns Dpd1 2 3P D Fdd0 1 2 3 4S p D F Gppp0 1 1 1 2 2 3S p P p D D FThe quantum number S is the absolute worth of the total electronspin abs(Σsi).Note: this S is not the exact same as the term S).Each electron has actually aspin of +/- 1/2. S is integral because that an even number of electrons,and fifty percent integral because that an strange number. S=0 for a closeup of the door shell.J represents the complete angular momentumof the atom the ion. That is the vector amount of L and also S.For ahydrogenic ion, L=0, S=1/2,and J=1/2. Because that morecomplex atom, J takes on the values L+S,L+S-1,..., abs(L-S), where absis the pure magnitude.For a given L, there are 2S+1 feasible values that J, unlessL<S, in which instance there space 2L+1 possible values ofJ.M, the Magnetic quantum number, bring away on worths of J, J-1,..., 0, ..., -J-1, -J.

## Spectroscopic Notation

The atom level is explained as n 2S+1LJwhere S, n, and also J room the quantum numbers definedabove, andL is the hatchet (S,P,D,F,G, etc). 2S+1 is themultiplicity.You may additionally see the level defined as n lx 2S+1LJ wherein l is the orbit of electron(s, p, d, etc.) and x is the numberof electron in the orbital (e.g., 1 or 2 for an s orbital, 1 to 6for a p orbital. n lxis the configurationof the outermost electrons.

## Terms, Configurations, and also Levels

The outermost electron in an atom or ion is the one that usually undergoestransitions, and so the state of that electron explains the state that the atomor ion. The configurationdescribes the n and also lvalues for all theelectrons in one atom. Because that example, the floor state of Boronhas a 1s22s22pconfiguration, with 2 electrons filling the n=1 level(l=0), 2 electron in the n=2, l=0 s orbital,and the fifthelectron start to populate the 2p orbital.The level is the collection of 2J+1 says with particular values that L,S, and also J.The differencein the energy in between two levels provides the wavelength or frequencyof an atomic transition.The term is the set of levels identified bya details S andL. The ground state that Boron has actually a 2P1/2 term.Closed shells constantly have a 1S0 term.Atoms whose external electrons have l=0,1,2,3,4 arereferred to together S, P, D, F, G terms,respectively (Note the an electron through l=0 is referred to as an s electron;lower caseterms describe individual electrons. For example, In the ground state, Boronhas 4 s electrons (2 in the n=1 level and also 2 in the n=2level) and one p electron. Theground state hatchet of the atom is P.Warning: the s in an s electron has nothingto execute with the quantum number s.).This is a carryover from beforehand spectroscopic nomenclature(sharp, principal, diffuse, and basic bands, through G followingF alphabetically) for alkali atoms, those with a closed shell of electron plusa single valence electron, such together Li, Na, K, Mg II, Ca II.

## Selection Rules

In complicated ions, there space an enormous number of possible transitions.Not every one of these feasible transitions are observed. This is due to the fact that some transitions are more likely 보다 others. Selection rules were landed on empirically to explain those alters in quantum numbers the were observed(permitted transitions) and those i beg your pardon did not (forbidden transitions).The basic selection rules, i beg your pardon strictly use only to straightforward configurationswhich obey strictly L-S coupling (In a simple atom or ion,L andS vector-sum to J.The level of L and S perform not influence eachother. This absence of communication is known as L-S coupling. In complexatoms or ions, levels of L and also S have the right to interact, causing abreakdown in L-S coupling (physicists have the right to be simply as illogical asatronomers in their nomenclature!). When this happens, L and also S are no longer interpretable in regards to angular momenta.),are: ΔL = 0, +/- 1 Δl = 1 ΔJ = 0, +/- 1, except that J=0 -> J=0 is forbidden. ΔS = 0 ΔM = 0, +/- 1, except that M=0 -> J=0 is forbidden if ΔJ=0.As the atom become much more complex,strict L-S coupling stops working to hold, and also theseselection effects come to be weaker.Permitted lines are those whose shift probability is high, i.e., whoseEinstein A worth is small, of order 108s-1. Forbiddenlines have small AUL values, of bespeak 1, due to the fact that they can not radiatein a dipole transition. Forbidden transitions are often feasible throughquadrupole or octopole transitions, which have low shift probabilities.Metastable lines have intermediate A values. A-1 offers the radiativelifetime ofthe excited state. Forbidden lines are characteristic that low density media,because in ~ high densities the time between collision is brief compared come theradiative lifetime, therefore collisional deexcitation is the leading process.

## Multiplets

Although the power levels in the Bohr version of Hydrogen depend only onthe principal quantum number n, in fact the degeneracy that thelevels are damaged by miscellaneous interactions in between the electron(s) and also thenucleus. Transitions occurring from a details nL hatchet (with a number ofvalues of J) to an additional term (which has multiple worths for J)give increase to a multiplet. For example, the Na i doublet (the FraunhoferD lines) in ~ 5890,5896 Angstroms arise from the32P - 32S transition. The32P term is break-up into 32P1/2 and 32P3/2levels. The power difference between these level with various Jvalues quantities to about 5.9 Angstroms.The multiplicity the a ax is offered by the valueof 2S+1. A termwith S=0 is a singlet term; S=1/2 is a doublet term; S=1 is a triplet term; S=3/2 is a quartet term, etc.Alkali steels (S=1/2) form doublets. Ion with2 electrons in the external shell, prefer He, Ca ns or Mg I, type singlets ortriplets. Multiplet separating increases v the degree of departure from strictL-S coupling.

### M Degeneracy and also the Zeeman Effect

The M degeneracy have the right to be broken by applications of a magnetic fieldB. This is the Zeeman effect. The regular Zeeman effect operates in a singlet state,and results in 3 lines. The lines with ΔM = 0, the πcomponents, room unshifted, and are polarized parallel to the magnetic field;the lines through ΔM = +/- 1, the σcomponents, are shifted by +/- 4.7 X 10-13g λ2B, whereby g is the Lande g factor,λ is the wavelength that the unshifted line in Angstroms, and Bis the stamin of the magnetic field in Gauss. The Lande g aspect is g = 1 + (J(J+1) + S(S+1) - L(L+1))/2J(J+1) The anomalous Zeeman effect operates on currently that room not singlets,and produces an ext than 3 components.

## Hyperfine Structure

Hyperfine structure arises from the coupling in between the magnetic momentof the electron and also the nuclear magnetic moment.For the quantum number Idefining the net nuclear spin (analagous to the net electron rotate S), you can construct one more quantum number F=I+J, i beg your pardon takeson values J-I,J-I+1, ... J+I-1,J+I. The 2S1/2 floor state the Hydrogen has actually J=1/2,I=1/2 (because the turn of the proton is 1/2), andF deserve to take on the worths 0 or 1. F=1 synchronizes to parallelspins for the proton and also electron; F=0 synchronizes to anti-parallelspins, and is the lower energy configuration. The energy differencecorresponds to a frequency of 1420 Mhz, or a wavelength of 21 cm. This is avery important line bromheads.tvphysically, for it permits united state to map the distributionof cold Hydrogen in ours galaxy and also the universe.

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### References

An great reference isAtomic Spectra and also Atomic Structureby Gerhard Herzberg, easily accessible as a Dover paperback.bromheads.tvphysics the the Sun, through Harold Zirin(Cambridge university Press, 1989) includes a an excellent discussion ofthis issue in chapter 5.See additionally chapters 2-4 the Optical bromheads.tvnomical Spectroscopy byC. R. Kitchin (Institute the Physics Publishing).Frederick M. Walter2 October 1998; update 26 September 2000